You cannot start counting 1,2,3,4,... ad infinitum and reach somewhere, anywhere. Infinity has neither a start or an end — Alkis Piskas

Then, counting (natural) numbers you can never reach infinity because that infility would be also a number, and infinity is not a real or natural number. — Alkis Piskas

Agreed (especially with "Infinity has neither a start or an end").

A set is a collection of objects (elements, members). I'm not sure if we can talk about an infinite set — Alkis Piskas

I don't believe we can talk about different infinite sets because it will lead to contradictions. But I do believe Infinity and Existence denote the same Entity. I see Infinity/Existence as the set of all existents. I see there being no end to the number of existents purely because the nature of Infinity/Existence allows for such possibilities.

All this raises questions about the infiniteness of the Universe, whether it started (created) from something or it always existed, etc. And, as I see it, since we don't have a proof that it is created from nothing, it must have always existed, even in the form of extremely high density and temperature, which at some point exploded (re: Big Bang), or in any other form. But I'm not the right person to talk about these things. — Alkis Piskas

It is clearly contradictory for something to come from nothing. And since I have heard some say that the universe is expanding, my view is that the universe is not infinite (if it's expanding, it's not infinite). But I do view Existence/Omnipresent as Infinite. I see Infinity as the reason for why an endless number of things can be imagined or thought about or experienced (dream or otherwise). Infinity has Infinite potential, therefore, an endless number of things can be imagined or thought about or experienced (we and our minds are wholly contingent on Existence. There is no non-Existence for us or our minds to draw anything from).

But Euclidian triangles don't exist in nature. — Lionino

To my knowledge they don't exist in our universe due to gravity. But I see our universe as just a part of Existence/Nature/Infinity. Something has to account for why we are aware that "the angles in a triangle add up to 180 degrees". To me, the nature of Existence/Infinity accounts for this awareness.

Existence can accommodate both perfect and imperfect triangles. We have experienced imperfect triangles (as in we have visually seen them), we have not experienced perfect triangles. But somehow, we have the knowledge that the angles in a triangle add up to 180 degrees. This is the awareness we have got in Existence and have gotten from Existence.

To me, the nature of Existence/Infinity accounts for this awareness. — Philosopher19

How?

Suppose that the universe has infinite space, and let's also say that there is an infinite number of particles in this space. For there to be space between the particles, would that not make space a bigger infinity than the infinite number of particles in the infinite space?

It would not! Are you familiar with injective, surjective, and bijective functions?

Suppose there are two sets of objects, A and B, whose size (cardinality) we wish to compare. That is, we want to know which is bigger (or equal): size(A) or size(B)? This is also written as card(A) and card(B).

If there is an injective function f : A → B mapping the objects of A into the objects of B, then for every distinct object in A one can find a distinct object in B. This entails that size(A) ≤ size(B).

If there is a surjective function f : A → B, then for every distinct object of B one can find a distinct object in A. This entails that size(A) ≥ size(B).

If there is a bijective function f : A → B, this just means that "f" is both injective and surjective, so that there is a one-to-one correspondence between elements of A and elements of B. This entails that size(A) = size(B).

This is the toolkit that defines the notion of size (cardinality) in set theory, and it must be used to compare sizes among sets, including infinite sets.

So take the positive natural numbers ℕ = { 1, 2, ··· }, which is infinite. Also take the non-zero integers ℤ = { ··· –2, –1, +1, +2, ··· }, which is also infinite. I have excluded zero for convenience.

Now, ℤ might seem bigger than ℕ. However, one can construct a bijective map between the two, proving that they have in fact the same size. There are many such possible maps. Any one of them suffices.
→ One such map begins by mapping all odd numbers in ℕ to the positive numbers in ℤ, like so: 1 maps to +1, 3 maps to +2, 5 maps to +3, and so on. The mapping rule is: 2k+1 → k+1 (where k starts from 0).
→ It continues by mapping all even numbers in ℕ to the negative numbers in ℤ, like so: 2 maps to –1, 4 maps to –2, 6 maps to –3. The mapping rule is: 2k → –k (where k starts from 1).
This covers all numbers both in ℕ and in ℤ, so they have the same size.

Infinite sets have this weird property where one can rearrange their items in many different ways, leading to surprising one-to-one correspondences. This is well illustrated by Hilbert's hotel.

In the case you provided, we could have 100 units of empty space for every 1 particle, but both would still be equinumerous, for there would still be a map between them. Just map the first 100 particles to the first 100 units of space, then the next 100 particles to the next 100 spaces, and so on. (This is an abstract mapping: you are not in fact shuffling particles around.) You will never run out of particles to map to some unit of space. Every particle will be mapped somewhere; every spatial unit will be designated a particle. So they might well be infinities of the same size.

This would be different if the space in question were continuous (like ℝ). Since particles are discrete (countable), they have the same cardinality as ℕ, which is strictly less than that of ℝ.

Suppose someone produces an axiom. Will it not be the case that that axiom will either be contradictory in relation to certain truths or consistent in relation to certain truths? Existence determines what is true and what is false. Whether any belief or axiom highlights truths or is contradictory to truth is determined by Existence/Truth. If not, there is no truth or semantics to work with to deduce further truths. — Philosopher19

I really don't think "truth" in this way is relevant. This is more of an issue of pragmatics, mathematics is a tool. You wouldn't say that one saw is more truly a saw than another saw, or on shovel is a more true shovel than another. So the axioms which are accepted, "which are bought", are the ones which mathematicians like to use. It may well be the case that existence determines truth, like you say, but that's not relevant to the selection of mathematical axioms.

I do believe we can bring "truth" into the picture in a different way though. Since mathematicians can choose to use whichever axioms they feel comfortable with, we can say that the axioms follow use. That means that they are a reflection of what mathematicians are doing. Therefore we can say that they are descriptive rather than prescriptive. The axioms do not give mathematicians rules for how to do things, because the mathematicians get to create and choose their own axioms. So the axioms simply provide a representation of what mathematicians are doing. Since they are descriptions, "truth" is to be found in how well the axioms represent what the mathematicians are actually doing. As an analogy, consider looking at a dictionary and judging how truthfully the definitions represent how people are actually using the words which are defined there.

I don't believe we can talk about different infinite sets because it will lead to contradictions. — Philosopher19

Right.

I do believe Infinity and Existence denote the same Entity. I see Infinity/Existence as the set of all existents. I see there being no end to the number of existents purely because the nature of Infinity/Existence allows for such possibilities. — Philosopher19

I see what you mean. Well, the words "exist" and "existence" can be used in different ways. And it can be used strictly (substantially, concretely) and loosely (insubstantially, abstactly). And I guess the second form applies to what you say above.

It is clearly contradictory for something to come from nothing. — Philosopher19

Right. Or, impossible. Yet, we stiil can use the expression "something from nothing" loosely or figuratively. But there are always some conditions (something) that allow the creation of some other thing (something). This is the universal law of cause and effect.

And since I have heard some say that the universe is expanding, my view is that the universe is not infinite (if it's expanding, it's not infinite). — Philosopher19

But doesn't an expanding universe mean that this process is infinite and thus the universe itself is limitless? It is not much different than if we consider the universe as being static, in which case it can also be infinite.

Well, that's why the infiniteness of the Universe is still debatable today! :smile:

But doesn't an expanding universe mean that this process is infinite and thus the universe itself is limitless? It is not much different than if we consider the universe as being static, in which case it can also be infinite. — Alkis Piskas

Infinitiness and expansion are independent. The Universe could be infinite and expanding; finite and expanding; infinite and static; and finite and static. All combinations are possible.

It is important to understand what it means to say that the Universe is expanding.

If the Universe is infinite, such expansion does not mean that the Universe is increasing in cardinality (set-theoretic size). Infinity is infinity (of a given size: aleph-0, aleph-1, aleph-2, and so on).

What the Universe's expansion means, whether it is infinite or not, is that its local energy density is decreasing. In other words, there is more spatial structure between each of its internal field excitations (particles, energy).

(Note that, if the Universe were infinite, its global density would remain constant. This is because global density would be calculated by dividing two infinite sets of the same cardinality.)

This touches a related point: the Universe is undergoing an internal expansion. It is not expanding *into* something. There is no space external to the Universe. The Universe is just acquiring more internal structure. The cosmological details are sure to be complicated and relevant (and I'm no cosmologist), but that is the general gist.

There is also a related point: the Universe could be finite but still unbounded, without an edge. It could just be twisted unto itself, as in a loop, like the surface of the Earth. On our planet, if you keep going North, you'll eventually just change hemispheres and start going South.

What the Universe's expansion means, whether it is infinite or not, is that its local energy density is decreasing. In other words, there is more spatial structure between each of its internal field excitations (particles, energy). — DanCoimbra

Hi Dan, I see you're new here, so welcome to this space.

I don't think it's proper to say that expansion means "more spatial structure" between internal field excitations, unless you are speaking of a "spatial structure" which is other than Einsteinian space-time.

How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached? — Philosopher19

Doesn't infinity mean endless? i.e. unreachable eternal continuation in concept?
If it was reachable, then it wouldn't be infinity. Any set or size would be unknowable, if it were infinity. Therefore talking about different size, set or number of infinity, is it not a nonsense?

Thanks for the warm welcome and the thoughtful reply

What is the proper interpretation of the cosmological constant Λ? I understand that it corresponds to a vacuum energy density, pervading all reality. Such energy is called dark energy, I gather. Since I'm sketchy on field theory, I don't know how this goes, but somehow this energy density produces a repulsive force beween any two objects in spacetime (within each other's lightcones?). Matter remains cohesive because Λ is very small compared to other forces, so that its effects really only show at an intergalactical scale (megaparsec).

Now, somehow this leads to the expansion of the Universe even in the case where the Universe is finite and bounded, which is a possibility considered by cosmologists. In this case, the Universe is increasing in total size, but not increasing *into* anywhere, so it becomes bigger because it has more internal spatial structure. This is what I meant. Why do you think this is incorrect?

One can talk about infinity conceptually, as one does in mathematics, without reference to its empirical verifiability.

When it comes to the empirical application of the concept of infinity, it is indeed reasonable to think that it is fundamentally unverifiable whether something is infinite. So we couldn't know whether spacetime is continous or discrete, because our measurements have finite resolution. The same would go for whether the Universe is infinite in extension or not.

However, humans can be quite ingenious, and we shouldn't rule out any possibility apriori, just from armchair thinking. Perhaps the supposition that the Universe is continuous rather than discrete has different consequences for our finite observations; I don't know. The same goes to the cosmological hypothesis where the Universe is infinite, which is thought to hold in case the Universe's matter density equals its dark matter density, a possibiity referred to as Ω = 1. (I'm pretty much quoting Wikipedia on the expansion of the Universe.)

Quite.a lot of theories of infinity,

Infinity by size, set, numbers, zero to infinity, infinity with no beginning and no end, infinity applied as mental constructs, infinity applied to physical matter, pairing of infinities, pairing infinite sets to finite sets, infinity as a simple concept of continuing without limit. Advanced math concepts of infinity......

Some of the advanced math theories are maybe just some mental showboating of things the math people can do with their brains.

Actually the original attempts are the most interesting from a philosophy perspective.

One can talk about infinity conceptually, as one does in mathematics, without reference to its empirical verifiability. — DanCoimbra

"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein.

Welcome to TPF~ :cool: :up:

Some of the advanced math theories are maybe just some mental showboating of things the math people can do with their brains. — Mark Nyquist

A person hasn't studied the pertinent mathematics, doesn't know anything about it, doesn't understand it. So their response to it is to say that it might be just a bunch of "mental showboating" anyway.

The 'showboating' is actually impressive to me so don't take it the worst way.

So your qualification is noted. But what you wrote originally naturally would be taken as a pejorative. "just mental showboating" would not ordinarily be understood as approbation.

Well, there are a lot of scenarios on the table. Let astronomers and cosmologists debate about them ...

Welcome to TPF! :clap:

Of course notions of infinity pertain in different areas of study. But just to bear in mind, the original post is a challenge to the idea that there are "different sizes" of infinity, which is primarily a set theoretic idea, and the post mentioned mathematics regarding that. So among philosophers, physicists, astronomers, cosmologists and theologists, we will still have mathematicians to consider.

That's kind of funny...just like it is

How? — Lionino

By Being. Existence just Is. It just is the case that triangles are triangular or that Existence is Infinite or that 1 plus 1 = 2. Or if you're interested in more on Existence, it just is the case that Existence indubitably exists and is Perfect. I won't go into detail with regards to how Existence is Perfect and indubitably exists. I'll just provide the link to the argument: http://godisallthatmatters.com/2021/05/03/the-image-of-god-the-true-cogito/

infinity is a concept considering continuity, not size. — Vaskane

Again, in mathematics, the concept is 'is infinite' as an adjective, not 'infinity' as a noun. And continuity is a different idea, while the idea of "size" is approached by the formulation of the idea of cardinality.

just like it is — Mark Nyquist

Nothing at all like it is.

As an analogy, consider looking at a dictionary and judging how truthfully the definitions represent how people are actually using the words which are defined there. — Metaphysician Undercover

Of course, it is possible, for example, for mathematicians to be using the label "infinity" to refer to a semantic that is different to the semantic of infinity. But from what I've seen of mathematicians, they either have no part for infinity, or they're using infinity wrong. I believe they're doing the latter which leads to the former (which I think is why I have heard it said before that "maths is incomplete")

It may well be the case that existence determines truth, like you say, but that's not relevant to the selection of mathematical axioms. — Metaphysician Undercover

I just think if mathematical axioms are to be selected, they have to be such that they do not lead to what is contradictory to Existence/Truth (or just semantics in general).

The axioms do not give mathematicians rules for how to do things, because the mathematicians get to create and choose their own axioms. So the axioms simply provide a representation of what mathematicians are doing. Since they are descriptions, "truth" is to be found in how well the axioms represent what the mathematicians are actually doing — Metaphysician Undercover

If a mathematician or a philosopher decides on an axiom or theory that requires belief in the following (or at least logically implies it or leads to it): Nothing can be the set of all things (which logically implies Existence is not the set of all existents), or one infinity is a different bigger than another (or is a different quantity than another), I believe that axiom or theory should be disregarded or at least viewed as contradictory to Existence/Truth (or at least contradictory to the semantic of infinity).

for mathematicians to be using the label "infinity" — Philosopher19

So your method of conversation is to ignore when someone informs you nearly a dozen times on a point:

In this context, mathematics doesn't use 'infinity' as a noun, as if there is an object named 'infinity', but rather 'is infinite' as an adjective to name a property. That distinction is crucial to understanding the subject matter.

from what I've seen of mathematicians — Philosopher19

What you've seen is what you've allowed yourself to see, which is virtually nothing about the actual mathematics you've not even bothered looked up.

I see what you mean. Well, the words "exist" and "existence" can be used in different ways. And it can be used strictly (substantially, concretely) and loosely (insubstantially, abstactly). And I guess the second form applies to what you say above. — Alkis Piskas

I don't mean to use Existence loosely/abstractly. By "Existence" I mean that which encompasses all things physical or otherwise (if otherwise is possible). So dreams (which some may view as non-physical) are clearly a part of Existence. The term universe seems limited to me in terms of accounting for all that exists (I cannot comfortably say something like "the universe has a space for dream worlds" whereas I can comfortably say "Existence has a space for all worlds including the universe and dreams"). To me, Existence/Infinity clearly fits the bill of 'encompasses all things/existents' whilst universe does not.

But doesn't an expanding universe mean that this process is infinite and thus the universe itself is limitless? It is not much different than if we consider the universe as being static, in which case it can also be infinite. — Alkis Piskas

If the universe is expanding, it is expanding in something. Some thing has to be Infinite to allow for the possibility/potentiality for the universe to forever expand. But it is also the case that even if the universe expands forever, it will not become infinite (this is not unlike me saying even if I count 1, 2, 3 ad infinitum, I will never reach infinity).

So to me, Infinity/Existence is the reason that something can expand forever or go on forever. As for the thing that expands (like the universe), it is a part of the Infinite. It is not itself infinite.

"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein.

Thanks for the welcome!

As regards Wittgenstein's remark, we use finite statements to fixate reference on infinite objects and work out their properties. There is no contradiction in that.

Here is a finite definition of an infinite set: "A given set S is infinite iff there exists a bijective function between S and a proper subset of S." Furthermore, such a bijective function can be stated finitely.

Here is an example. Take the set of natural numbers ℕ = { 0, 1, ··· }. Now take a proper subset of ℕ containing only even the numbers, ℙ = { 0 , 2 , ··· }. These two are equinumerous because there is a bijective function f : ℕ → ℙ, given by f(n) = 2n.

The proof that "f" is bijective is finite. So is the proof that ℙ is a proper subset of ℕ.

The above post is correct and mentions a crucial point. Therefore, it will be of no interest to too many people.

When it comes to the empirical application of the concept of infinity, it is indeed reasonable to think that it is fundamentally unverifiable whether something is infinite — DanCoimbra

I get your point with regards to empirically verifying infinity, but I believe the a priori is superior to the a posteriori in that whatever observation we make (scientific or otherwise), has to be interpreted in line with the dictates of pure reason. It also has to be such that it does not contradict the semantics that we are aware of (for example, we must not have a theory that amounts to saying or logically implying that triangles don't have three sides because that contradicts the semantic of triangle) .

If a scientist says something like "I have observed something pop in and out of Existence" because it may have looked that way to him, we have to reject him because 'something popping in and out of Existence' is clearly contradictory. Non-Existence does not Exist for something to pop into or come out of. Things can be turned on and off but this is not the same as things popping in and out of Existence.

In the context of the discussion and the differing opinions I might have been suggesting a while back that the mathematicians here should (occasionally)take their metaphorical pen from the mathematics page to a brain theory of mathematics page.

If you understand that our brains are churning out stand alone theories that work fine in a certain context but don't all work together in every context you will better understand why we disagree.

Is that reasonable? Keep doing what you are doing.

Not 'infinity' as a noun, as if there is an object named 'infinity', but rather 'is infinite' as an adjective to name a property. — TonesInDeepFreeze

It makes no difference. Existence is Infinity (here it is a noun). Existence is Infinite (here it is an adjective). You cannot become Infinite (adjective) even if you expand forever. You are not Infinity (noun) if you are not Infinite (adjective).

You keep saying I ignore your points, but rightly or wrongly, I also think you have not read or considered what I've written with sufficient attention to detail.

What you've seen is what you've allowed yourself to see, which is virtually nothing about the actual mathematics you've not even bothered looked up. — TonesInDeepFreeze

I am not claiming to have seen everything. But (again, rightly or wrongly) I think I've seen enough to say:

But from what I've seen of mathematicians, they either have no part for infinity, or they're using infinity wrong. I believe they're doing the latter which leads to the former (which I think is why I have heard it said before that "maths is incomplete") — Philosopher19

In any case, in the event that you have made good points and I have failed to give them the right amount of attention, I apologise. I do think that I am being sincere and honest in this discussion (as well as not closed-minded).